Square Root of 189

√189 at a glance

Exact value

3√21

Decimal (10 places)

13.7477270849

Rounded

13.7 · 13.75 · 13.748

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.747727

Prime factorization

3³ × 7

Cube root

5.738794

How to simplify √189

Look for the largest perfect square that divides 189. Here it is 9 (3²), because 189 = 9 × 21 and 21 has no square factor left:

The prime factorization tells the same story: 189 = 3³ × 7. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 × 7 stays inside.

Check: (3√21)² = 3² × 21 = 9 × 21 = 189. As a decimal, 3√21 = 3 × 4.582575695 ≈ 13.7477270849.

Where √189 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √189 lies between 13 and 14. 189 is 20 above 169 and 7 below 196, so the root is closer to 14.

• Straight line between 169 and 196: 13.7407 (0.05% low)
• Tangent from 13, i.e. 13 + 20 ÷ 26: 13.7692 (0.16% high)
• Tangent from 14, i.e. 14 − 7 ÷ 28: 13.7500 (0.02% high)

For √189 the tangent at 14 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 189 is just 7 below 196.

Finding √189 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 189: following the tangent line down to zero simplifies to averaging x with 189 ÷ x.

Start from the nearest whole number, 14 (14² = 196):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √189 = 13.7477270849 to every decimal shown.

√189 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √189 the pattern is [13; 1, 2, 1, 26] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √189 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 189y² = 1. Its smallest solution in positive whole numbers is x = 55, y = 4.

√189 in geometry and everyday measurements

• A square room or garden bed covering 189 square feet measures about 13.75 ft (13 ft 9 in) along each wall.
• 189 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √189 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 13 box, because 2² + 4² + 13² = 189.
• Since √189 = 3√21, a length of √189 is exactly 3 copies of the length √21 laid end to end.

Square roots near √189 and related roots

• The cube root of 189 is about 5.738794.
• Four times the radicand doubles the root: √756 = 2 × √189 ≈ 27.495454.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.